Conformal Mappings of Mixed Generalized Quasi-Einstein Manifolds Admitting Special Vector Fields

E extracta mathematicae Vol. 32, N´um.2, 255 – 273 (2017)

S. Dey ∗, A. Bhattacharyya

Department of Mathematics, Jadavpur University Kolkata-700032, India [email protected] [email protected]

Presented by Manuel de Le´on Received October 28, 2016

Abstract: It is known that Einstein manifolds form a natural subclass of the class of quasi- Einstein manifolds and plays an important role in geometry as well as in general theory of relativity. In this work, we investigate conformal mapping of mixed generalized quasi- Einstein manifolds, considering a conformal mapping between two mixed generalized quasi- Einstein manifolds Vn and V¯n. We also find some properties of this transformation from Vn to V¯n and some theorems are proved. Considering this mapping, we peruse some properties of these manifolds. Later, we also study some special vector fields under these mapping on this manifolds and some theorems about them are proved. Key words: mixed generalized quasi-Einstein Manifolds, φ(Ric)-vector field, concircular vector field, Codazzi tensor, conformal mapping, conharmonic mapping, σ(Ric)-vector field, ν(Ric)-vector field. AMS Subject Class. (2010): 53C25, 53C15.

1. Introduction

The notion of quasi-Einstein manifold was introduced by M.C. Chaki and R.K. Maity [5]. A non-ﬂat Riemannian manifold (M n, g), (n ≥ 3) is a quasi- Einstein manifold if its Ricci tensor S satisﬁes the condition

S(X,Y ) = ag(X,Y ) + bϕ(X)ϕ(Y ) and is not identically zero, where a, b are scalars, b ≠ 0 and ϕ is a non-zero 1-form such that

g(X,U) = ϕ(X) , for all X ∈ χ(M) ,

U being a unit vector ﬁeld.

∗ First author supported by DST/INSPIRE Fellowship/2013/1041, Government of India.

255 256 s. dey, a. bhattacharyya

Here a and b are called the associated scalars, ϕ is called the associated 1-form and U is called the generator of the manifold. Such an n-dimensional manifold denoted by (QE)n. As a generalization of quasi-Einstein manifold in [7], U.C. De and G.C. Ghosh defined the generalized quasi-Einstein manifold. A non-flat Rieman- nian manifold is called generalized quasi-Einstein manifold if its Ricci-tensor is non-zero and satisfies the condition

S(X,Y ) = ag(X,Y ) + bϕ(X)ϕ(Y ) + cψ(X)ψ(Y ) , where a, b and c are non-zero scalars and ϕ, ψ are two 1-forms such that

g(X,U) = ϕ(X) and g(X,V ) = ψ(X) ,

U and V being unit vectors which are orthogonal, i.e.,

g(U, V ) = 0 .

The vector fields U and V are called the generators of the manifold. This type of manifold will be denoted by G(QE)n. The notion of mixed generalized quasi Einstein manifold was introduced by A. Bhattacharya, T. De and D. Debnath in their paper [2]. A non-flat Riemannian manifold is called mixed generalized quasi-Einstein manifold if its Ricci-tensor is non-zero and satisfies the condition

S(X,Y ) = ag(X,Y ) + bϕ(X)ϕ(Y ) + cψ(X)ψ(Y ) + d[ϕ(X)ψ(Y ) + ϕ(Y )ψ(X)] , (1.1) where a, b, c and d are non-zero scalars and ϕ, ψ are two 1-forms such that

g(X,U) = ϕ(X) and g(X,V ) = ψ(X) , (1.2)

U and V being unit vectors which are orthogonal, i.e.,

g(U, V ) = 0 .

The vector ﬁelds U and V are called the generators of the manifold. This type of manifold will be denoted by MG(QE)n. Putting X = Y = ei in (1.1), we get

r = na + b + c . (1.3) conformal mappings 257

Here r is the scalar curvature of MG(QE)n where {ei}, i = 1, 2, . . . , n is an orthonormal basis of the tangent space at each point of the manifold. Quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces of semi Euclidean spaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. So quasi-Einstein manifolds have some importance in the general theory of relativity. One of the important concepts of Riemannian Geometry is conformal mapping. Conformal mappings of Riemannian manifolds (or semi-Riemannian manifolds) have been investigated by many authors. In general relativity, conformal mappings are important since they preserve the causal structure up to time orientation and light-like geodesics up to parametrization [13]. The existence of conformal mappings of Riemannian manifolds onto Einstein manifolds have been studied by Brinkmann [3], Mikeˇs,Gavrilchenko, Gladysheva [14] and others. Also, conformal mappings between two Einstein manifolds have been examined by Brinkmann. What is more, the problem of finding the invariants under a particular type of mapping is an important and active research topic. In particular, Gover and Nurowski [9] obtained the polynomial conformal invariants, the vanishing of which is a necessary and sufficient for an n-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold, and some of the invariants have certain practical significance in physics, such as quantum field theory [4], general relativity [1]. Motivated by the above studies the present paper provides conformal mapping on MG(QE)n admitting special vector fields. In the second section, we study conformal mapping of two mixed generalized quasi-Einstein manifolds Vn and V¯n. We also find some properties of these transformation from Vn to V¯n and some theorems are proved. Third section deals with conformal mapping on MG(QE)n admitting special vectors fields and in the final section we give an example of MG(QE)n.

2. Conformal Mapping of two Mixed Generalized Quasi-Einstein Manifolds

In this section, we suppose that Vn and V¯n,(n ≥ 3) are two mixed generalized quasi-Einstein manifolds with metrics g andg ¯ , respectively.

Definition 1. A conformal mapping is a diﬀeomorphism of Vn onto V¯n such that g¯ = e2σg , (2.1) 258 s. dey, a. bhattacharyya

where σ is a function on Vn. If σ is constant, then it is called a homothetic mapping.

In local coordinates, (2.1) is written as

2σ ij 2σ ij g¯ij(x) = e (x)gij(x) , g¯ (x) = e (x)g (x) , (2.2)

Besides those equations, we have the Christoﬀel symbols, the components of the curvature tensor, the Ricci tensor, and the scalar curvature, respectively

¯h h h h − h Γij = Γij + δi σj + δj σi σ gij , ¯h h h − h hα − △ h − h Rijk = Rijk + δk σij δj σik + g (σαkgij σαjgik + 1σ(δk gij δj gik) ,

S¯ij = Sij + (n − 2)σij + (△2σ + (n − 2)△1σ)gij , (2.3) −2σ r¯ = e (r + 2(n − 1)△2σ + (n − 1)(n − 2)△1σ) , (2.4) where ∂σ S = Rα , r = S gαβ , σ = = ∇ σ , σh = σ gαh (2.5) ij ijα αβ i ∂xi i α and σij = ∇j∇iσ − ∇iσ∇jσ . (2.6)

∇1σ and ∇2σ are the ﬁrst and the second Beltrami’s symbols which are de- termined by

αβ αβ △1σ = g ∇ασ∇βσ , △2σ = g ∇β∇ασ , (2.7) where ∇ is the covariant derivative according to the Riemannian connection in Vn. We denote the objects of space conformally corresponding to Vn by a bar, i.e., V¯n. If Vn is a MG(QE)n, then, from (1.1), (2.2) and (2.3), we have [ ] [ ] ¯bϕ¯ ϕ¯ +c ¯ψ¯ ψ¯ + d¯ ϕ¯ ψ¯ + ϕ¯ ψ¯ = bϕ ϕ + cψ ψ + d ϕ ψ + ϕ ψ i j i j i j j j i j i j { i j j i + (n − 2)σij + △2σ + (n − 2)△1σ } 2σ + a − ae¯ gij . (2.8)

Definition 2. A vector ﬁeld ξ in a Riemannian manifold M is called torse-forming if it satisﬁes the condition

∇X ξ = ρX + λ(X)ξ , conformal mappings 259 where ξ ∈ χ(M), λ(X) is a linear form and ρ is a function, [16]. In the local transcription, this reads ∇ h h h iξ = ρδi + ξ λi , (2.9) h h ξ and λi are the components of ξ and ϕ, δi is the Kronecker symbol. A torse-forming vector field ξ is called recurrent if ρ = 0; concircular if the form λi is a gradient covector, i.e., there is a function ϑ(x) such that λ = dϑ(x); convergent, if it is concircular and ρ = const. exp(ϑ). Therefore, recurrent vector fields are characterized by the following equation from (2.9) ∇iξj = λiξj . Also, from Definition 2., for a concircular vector field ξ, we get

∇iξj = ρigij (2.10) for all X,Y ∈ χ(M). A Riemannian space with a concircular vector ﬁeld is called equidistant, [15, 16]. Conformal mappings of Riemannian spaces (or semi-Riemannian spaces) have been studied by many authors, [3, 6, 8, 14]. In this section, we investigate the conformal mappings of mixed generalized quasi-Einstein manifolds preserving the associated 1-forms ϕ(X) and ψ(X).

Theorem 1. If Vn admits a conformal mapping preserving the associated 1-forms ϕ(X) and ψ(X) and the associated scalars b and c, then Vn is an equidistant manifold.

Proof. Suppose that Vn admits a conformal mapping preserving the associated 1-forms ϕ(X) and ψ(X) and the associated scalars b and c. Using (2.8), we obtain 2σ (n − 2)σij + (β + a − ae¯ )gij = 0 , where 2σ β = △2σ + (n − 2)△1σ + a − ae¯ . In this case, we get σij = αgij , (2.11) where 1 α = (¯ae2σ − a − β) n − 2 is a function. Putting ξ = − exp(−σ) and using (2.5), (2.6), (2.10) and (2.11), we get that Vn is an equidistant manifold. Hence, the proof is complete. 260 s. dey, a. bhattacharyya

Theorem 2. An equidistant manifold Vn admits a conformal mapping preserving the associated 1-forms ϕ(X) and ψ(X) if the associated scalars a¯, ¯b and c¯ satisfy both of the conditions d¯= d , c¯ = c , ¯b = b , a¯ = e−2σ(a + γ) , where (n − 1) [ ] γ = 2△ σ + (n − 2)△ σ . n 2 1

Proof. Suppose that Vn is an equidistant manifold. Then, there exists a concircular vector ﬁeld ξ satisfying the condition (2.10), that is, we have

∇jξi = ρgij , (2.12) where ξi = ∇iξ. Putting σ = − ln(ξ(X)) and using the condition (2.3), we obtain S¯ij = Sij + γgij , where (n − 1) [ ] γ = 2△ σ + (n − 2)△ σ . n 2 1 Considering (1.1) in (2.12) and using (2.2), we get [ ] ae¯ 2σg + ¯bϕ¯ ϕ¯ +c ¯ψ¯ ψ¯ + d¯ ϕ¯ ψ¯ + ϕ¯ ψ¯ = (a + γ)g + bϕ ϕ ij i j i j i j j [ j ] ij i j + cψiψj + d ϕiψj + ϕjψi . (2.13) If we take d¯= d,c ¯ = c, ¯b = b anda ¯ = e−2σ(a + γ), then from (2.13) we get

ϕ¯iϕ¯j = ϕiϕj ,

ψ¯iψ¯j = ψiψj ,

ϕ¯iψ¯j = ϕiψj and ϕ¯jψ¯i = ϕjψi . These completes the proof. The conharmonic transformation is a conformal transformation preserving the harmonicity of a certain function. If the conformal mapping is also conharmonic, then we have [11], 1 ∇ σi + (n − 2)σiσ = 0 . (2.14) i 2 i conformal mappings 261

Theorem 3. Let Vn be a conformal mapping with preservation of the associated 1-forms ϕ(X) and ψ(X) and the associated scalars b and c. A necessary and suﬃcient condition for this conformal mapping to be conharmonic is that the associated scalar a¯ be transformed by a¯ = e−2σa, ¯b = e−2σb, c¯ = e−2σc.

Proof. We consider a conformal mapping of quasi-Einstein manifolds Vn and V¯n. Then, we have from (1.1) and (2.3), we have [ ] [ ] ¯bϕ¯ ϕ¯ +c ¯ψ¯ ψ¯ + d¯ ϕ¯ ψ¯ + ϕ¯ ψ¯ = bϕ ϕ + cψ ψ + d ϕ ψ + ϕ ψ i j i j i j j j i j i j { i j j i + (n − 2)σij + △2σ + (n − 2)△1σ } 2σ + a − ae¯ gij . (2.15)

Multiplying (2.15) by gij and using (1.2), (2.1), (2.6) and (2.7), it can be seen that the following relation is satisﬁed

−2σ na¯ + ¯b +c ¯ = e [na + b + c + 2(n − 1)△2σ

+ (n − 1)(n − 2)△1σ] . (2.16)

If the conformal mapping is also conharmonic, then we have from (2.7) and (2.14)

2△2σ + (n − 2)△1 = 0 . (2.17) Considering (2.17) in (2.16), it is found that

na¯ + ¯b +c ¯ = nae−2σ + be−2σ + ce−2σ . (2.18)

From the equation (2.18), it can be seen that the associated scalars are transformed by a¯ = e−2σa , ¯b = e−2σb , c¯ = e−2σc . (2.19)

Conversely, if the associated scalars of the manifolds are transformed by (2.19), then we have from (2.16),

2(n − 1)△2σ + (n − 1)(n − 2)△1σ = 0 and so, we get the relation (2.14). Thus, the conformal mapping is also conharmonic. This completes the proof. 262 s. dey, a. bhattacharyya

Definition 3. A φ(Ric)-vector field is a vector field on an n-dimensional Riemannian manifold (M, g) and Levi-Civita connection ∇, which satisfies the condition ∇φ = µRic , (2.20) where µ is a constant and Ric is the Ricci tensor [10]. When (M, g) is an Einstein space, the vector field φ is concircular. Moreover, when µ = 0, the vector field φ is covariantly constant. In local coordinates, (2.17) can be written as ∇jφi = µSij , α where Sij denote the components of the Ricci tensor and φi = φ giα.

Suppose that Vn admits a σ(Ric)-vector ﬁeld. Then, we have

∇jσi = µSij , (2.21) where µ is a constant. Now, we can state the following theorem:

Theorem 4. Let us consider the conformal mapping (2.1) of a MG(QE)n Vn with constant associated scalars being also conharmonic with the σ(Ric)- vector ﬁeld. A necessary and suﬃcient condition for the length of σ to be constant is that the sum of the associated scalars b and c of Vn be constant.

Proof. We consider that the conformal mapping (2.1) of a MG(QE)n Vn admitting a σ(Ric)-vector ﬁeld is also conharmonic. In this case, comparing (2.14) and (2.21), we get (2 − n) r = σiσ , (2.22) 2µ i where r is the scalar curvature of Vn. If Vn is of the constant associated scalars, from (1.1) and (2.22), we ﬁnd [ ] (2 − n) b + c = σiσ − na . 2µ i

i If the length of σ is constant, then σ σj = c1, where c1 is a constant. Thus, we can see that b + c is constant. The converse is also true. Hence, the proof is complete. In [10], it was shown that Riemannian manifolds with a φ(Ric)-vector ﬁeld of constant length have constant scalar curvature. The converse of this theorem is also true. We need the following theorem [12], for later use. conformal mappings 263

Theorem 5. Let Vn be a Riemannian manifold with constant scalar curvature. If Vn admits a φ(Ric)-vector ﬁeld, then the length of φ is constant.

Now, we consider a MG(QE)n admitting the generator vector ﬁeld U as a ϕ(Ric)-vector ﬁeld. Then we have from (2.20)

∇jϕi = µSij and ∇jψi = µSij , (2.23) where µ is a constant. Then, we give the following theorem:

Theorem 6. In a MG(QE)n, if the vector fields U and V corresponding to the 1-forms ϕ and ψ are ϕ(Ric)-vector field and ψ(Ric)-vector field , then U and V are covariantly constant.

Proof. We consider a MG(QE)n whose generator vector ﬁeld is a ϕ(Ric)- vector ﬁeld. Putting (1.1) in (2.23), we obtain [ ] ∇jϕi = µSij = µ agij + bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} . (2.24)

Multiplying (2.24) by ϕi and using the condition g(U, U) = 1, it can be seen that i µSijϕ = µ(a + b)ϕj + µdψj = 0 . (2.25) Now multiplying (2.25) by ϕj, we get

µ(a + b) = 0 . (2.26)

As µ is non-zero, so from (2.26), we get

a = −b . (2.27)

Similarly putting (1.1) in (2.23), we obtain [ ] µSij = ∇jψi = µ agij + bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} . (2.28)

Again multiplying (2.28) by ψi, it can be seen that

i µSijψ = µ(a + c)ϕj + dµϕj = 0 . (2.29)

Now multiplying (2.29) by ψj, we get

µ(a + c) = 0 . (2.30) 264 s. dey, a. bhattacharyya

Similarly from (2.30), we obtain

a = −c . (2.31)

By the aid of (1.1), (2.27) and (2.31), we obtain

Sij = a[gij − (ϕiϕj + ψiψj)] + d{ϕiψj + ϕjψi} . (2.32)

i Otherwise taking the covariant derivative of the expression Sijϕ and using (2.24), we obtain ∇ i i ( kSij)ϕ + µSijSk = 0 . (2.33) Multiplying (2.33) by gjk, we obtain

∇ k i ij ( kSi )ϕ + µSijS = 0 , (2.34)

ij jk i where S = g Sk. It was shown, [10], that Riemannian manifolds with a ϕ(Ric) vector field of constant length have constant scalar curvature. Since the generator U is a unit vector field and it is also a ϕ(Ric) vector field, the scalar curvature of the manifold is constant. In this case, using the contracted second Bianchi identity and considering that the scalar curvature of the manifold is constant, it is obtained that 1 ∇ Sk = ∇ r = 0 . (2.35) k i 2 i Using (2.34) and (2.35) and assuming that µ is a non-zero constant, we obtain

ij SijS = 0 . (2.36)

By the aid of (2.32) and (2.36) it follows that

(n − 2)a2 + 2d2 = 0 . (2.37)

Since n > 2, from (2.37), it is seen that a and d must be zero, that is, a = c = 0 = d. But, in this case, from (2.32) we get that the Ricci tensor vanishes which is a contradiction to the hypothesis. Therefore, the constant µ must be zero and so, the generator vector field U is covariantly constant. Similalry, if we take ψ(Ric)-vector field, then we can show that the generator vector field V is also covariantly constant. This completes the proof.

Now we prove the following theorem: conformal mappings 265

Theorem 7. In a MG(QE)n admits a φ(Ric)-vector ﬁeld and ν(Ric)- vector ﬁeld with constant length, then either ϕi, ψi and φi are coplanar or the Ricci tensor of the manifold reduces to the following form

Sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi}

i if ψi and φ are orthogonal to each other and ψi, ϕi and νi are coplanar or the Ricci tensor of the manifold reduces to the following form

Sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi}

i if ϕi and ν are orthogonal to each other.

Proof. We assume that MG(QE)n admits a φ(Ric)-vector ﬁeld and ν(Ric)-vector ﬁeld with constant length. Then, we have

i i φiφ = p(say) and νiν = q(say) , (2.38) where c is a constant. Taking the covariant derivative of the condition (2.38), using the equation (2.23) and considering µ as a non-zero constant (that is φ is proper φ(Ric)-vector ﬁeld), it follows that

i Sikφ = 0 . (2.39)

By the aid of (1.1) and (2.39), we get { } i i i i aφk + b(φ ϕi)ϕk + c(ψiφ )ψk + d ϕiψkφ + ϕkψiφ = 0 . (2.40)

Multiplying (2.40) by ϕk and using (1.2), it is obtained that

k i (a + b)φkϕ + dψiφ = 0 . (2.41)

i If we take ψi and φ are orthogonal to each other, then from (2.41), we obtain

k (a + b)φkϕ = 0 .

k So either φkϕ = 0 which gives from (2.40) that

aφk = 0 .

So, we get a = 0 and so, the Ricci tensor of the manifold reduces to the form

Sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} 266 s. dey, a. bhattacharyya

k or φkϕ ≠ 0 which gives from (2.28) that a = −b . (2.42)

Again taking the covariant derivative of the condition (2.38), using the equation (2.23) and considering µ as a non-zero constant (that is ν is proper ν(Ric)-vector ﬁeld), it follows that

i Sikν = 0 . (2.43) Using the equation (1.1) and (2.43), we get

i i i i aνk + b(ν ϕi)ϕk + c(ψiν )ψk + d{ϕiψkν + ϕkψiν } = 0 . (2.44) Multiplying (2.44) by ψk and using (1.2), it is obtained that

k i (a + c)νkψ + dϕiν = 0 . (2.45)

i If we take ϕi and ν are orthogonal to each other, then from (2.45), we obtain

k (a + c)νkψ = 0 .

k So either νkψ = 0 which gives from (2.44) that

aνk = 0 . Thus, we get a=0. and so, the Ricci tensor of the manifold reduces to the form Sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} k or νkψ ≠ 0 which gives from (2.45) that a = −c . (2.46)

Since b ≠ 0 and c ≠ 0 then a ≠ 0 and using the equation (2.40), (2.42) and (2.46), we obtain that

i i i φk = (φ ϕi)ϕk + (ψiφ − dϕiφ )ψk . (2.47)

So from (2.47) we say that φk, ϕk and ψk are coplanar. Again from (2.44), we obtain

i i i νk = (ν ϕi)ϕk + (ψiν − dϕiν )ψk , (2.48) i.e., νk, ϕk and ψk are also coplanar. conformal mappings 267

Corollary 1. If a MG(QE)n admits φ(Ric)-vector field and ν(Ric)- vector field with constant length which is not orthogonal to the generators, then the associated scalars of the manifold must be constants and the vector fields φ and ν are covariantly constant.

Proof. As it has been alluded before that a Riemannian manifold admitting a φ(Ric)-vector field and ν(Ric)-vector field with constant length has constant scalar curvature. Besides, under the assumptions and from Theorem 7., we obtain that the associated scalars of MG(QE)n are connected by a = −b and a = −c, and from (1.3), we obtain r = (n − 2)a . (2.49) Since the scalar curvature of the manifold is constant, in this case, from (1.3) and (2.49), we see that the associated scalars of the manifold are constants. For the second part, multiplying the equation (2.47) by φk and using i i (2.38), it can be ocular that φ ϕi is a constant as ψiφ is also constant. So, (2.47) displays that the generator vector field U is also a ϕ(Ric)-vector field. In this case, U must be covariantly constant by Theorem 6. Again, multiplying k i i (2.48) by ν and using (2.38), it can be seen that ψiν is a constant as ν ϕi is also constant. Now due to the coplanarity of φ, U and V , φ is covariantly constant. Similarly, due to the coplanarity of ν, U and V , ν is also covariantly constant. Hence the proof is completed.

3. Conformal mapping of MG(QE)n admitting special vector fields

Definition 4. A symmetric tensor field T of type (0,2) on a Riemannian manifold (M, g) is said to be a Codazzi tensor if it satisfies the following condition (∇X T )(Y,Z) = (∇Y T )(X,Z) (3.1) for arbitrary vector fields X,Y and Z. ′ Now, we assume that the Ricci tensors S and S of the MG(QE)n are Co- dazzi tensors with respect to the Levi-Civita connections r′ and r, respectively. Then, from (3.1), we have the following relations

∇¯ kS¯ij = ∇¯ jS¯ik (3.2) and ∇kSij = ∇jSik . (3.3) 268 s. dey, a. bhattacharyya

On the other hand, if the Ricci tensor of the manifold is a Codazzi tensor, then from the second Bianchi identity, it can be seen that the scalar curvature is constant. According to our assumptions, the scalar curvatures r′ and r of the quasi-Einstein manifolds are constants. So, we state and prove the following theorems.

2σ Theorem 8. Let us consider a conformal mapping g¯ = ge of MG(QE)n whose Ricci tensors are Codazzi type. If the vector ﬁeld generated by the 1-form σ is a σ(Ric)-vector ﬁeld, then either this conformal mapping is homothetic or the relation (2 − n)(n − 1)c′ − (na + b + c) µ = 2(n − 1)(na + b + c)

′ ∂σ is satisﬁed where c is the square of the length of σi = ∂xi = ∂iσ and and µ denotes the constant corresponding to the σ(Ric)-vector ﬁeld.

Proof. Suppose that the Ricci tensors of Vn and V¯n are Codazzi tensors and suppose thatg ¯ = ge2σ is a conformal mapping with a σ(Ric)-vector ﬁeld. By using the second Bianchi identity, it can be seen that the scalar curvatures r andr ¯ are constants. Since r is constant, then the length of σi is constant by Theorem 5., (and r ≠ 0 which can be seen from Theorem 7 and Corollary 2.1) and so we have the condition

i ′ σiσ = c , (3.4) where c′ is a constant. If we assume that the vector ﬁeld generated by the 1-form σ in the conformal mapping (2.1) is a σ(Ric)-vector ﬁeld, we get

∇jσi = µSij , (3.5) where µ is a constant. Using (2.7), (3.4) and (3.5), we have the following relations ′ △2σ = µr , △1σ = c (3.6) and so, △1σ and △2σ are constants. Using the relations (3.6) in (2.4), we ﬁnd

r¯ = e−2σB, (3.7) where r,r ¯ and B = r + 2(n − 1)µr + (n − 1)(n − 2)c are constants. In this case, ifr ¯ is non-zero then we get from (3.7) that B is non-zero and so, e−2σ conformal mappings 269 is constant. Thus, σ is constant. Therefore, this mapping is homothetic. Ifr ¯ is zero then B must be zero. So we obtain using (1.3) (2 − n)(n − 1)c′ − (na + b + c) µ = . 2(n − 1)(na + b + c) This completes the proof.

Next we consider a conformal mapping between two MG(QE)n admitting a concircular vector ﬁeld σi.

2σ Theorem 9. Let us consider a conformal mapping g¯ = ge of MG(QE)n whose Ricci tensors are Codazzi type. If σi is a concircular vector field, then either (i) σ is orthogonal to ϕh, or k k [ ] c b − − + (n − 2)△1σ (ii) the function ρ is found as ρ = n 1 , n + 2 and (iii) σ is orthogonal to ψh, or k k [ ] b c − − + (n − 2)△1σ (iv) the function ρ is found as ρ = n 1 , n + 2 where ϕi, ψi denote the components of the vector field associated 1-form ϕ ∂σ and ψ, σi = ∂xi = ∂iσ, b, c are the associated scalar of Vn and ρ denotes the function corresponding to the concircular vector field.

Proof. Let the Ricci tensors of Vn and V¯n be Codazzi tensors and σi be a concircular vector ﬁeld. In this case, we have from (2.10)

∇jσi = ρgij , (3.8) where ρ is a function. Taking the covariant derivative of S¯ij and using (2.4), it can be obtained that

(∇¯ S¯ij) = ∇Sij + (n − 2)∇kσij + ∂k(△2σ + (n − 2)△1σ)gij − 2σkSij

− σiSjk − σiSik − 2(△2σ + (n − 2)△1σ)gijσk h h + σ (Sihgjk + Shjgik + (n − 2)(σ σhjgik h + σ σihgjk − 2σkσij − σiσkj − σjσik). (3.9) 270 s. dey, a. bhattacharyya

Changing the indices j and k in (3.9) and subtracting the last equation from (3.8) and using (2.6), (3.2), (3.3) and (3.8), it can be seen that

2(n − 1)(ρkgij − ρjgik) + [(n − 2)△1σ + (n + 2)ρ](σjgik − σkgij) h h + σjSik − σkSij + σ Shjgik − σ Shkgij = 0 . (3.10)

Multiplying (3.10) by gij, it is obtained that

2 2(n − 1) ρk + [(n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ − r]σk h + (2 − n)σ Shk = 0 . (3.11)

On the other hand, we have from the Ricci identity and the equation (3.8) α − σαRijk = ρkgij ρjgik , (3.12)

α Rijk denote the components of the curvature tensor. Multiplying (3.12) by gij, we obtain α − σαSk = (n 1)ρk . (3.13)

Substituting ρk obtained from (3.13) in (3.11), it can be obtained that

h nσ Shk + [(n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ − r]σk = 0 . (3.14)

Considering (1.1) in (3.14) and using (1.3), we get [ ] [ nσh bϕ ϕ + cψ ψ + d{ϕ ψ + ϕ ψ } + (n − 2)(1 − n)△ σ h k h k i j j i ] 1 + (n + 2)(1 − n)ρ − b − c σk = 0 . (3.15)

Multiplying (3.15) by ϕk and using (1.2), we obtain [ (n − 1)b − c + (n − 2)(1 − n)△1σ ] k h + (n + 2)(1 − n)ρ σ ϕk + ndσ ψh = 0 . (3.16)

Multiplying (3.16) by ϕh, we obtain [ ] − − − − △ − k h (n 1)b c + (n 2)(1 n) 1σ + (n + 2)(1 n)ρ σ ϕk = 0 . (3.17) Again multiplying (3.15) by ψk and using (1.2), we obtain [ ] k h (n−1)c−b+(n−2)(1−n)△1σ +(n+2)(1−n)ρ σ ψk +ndσ ϕh = 0 . (3.18) conformal mappings 271

Multiplying (3.18) by ψh, we obtain [ ] − − − − △ − k h (n 1)c b + (n 2)(1 n) 1σ + (n + 2)(1 n)ρ σ ψk = 0 . (3.19)

From (3.17), we see that either

k h σ ϕk = 0 or (n − 1)b − c + (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ = 0 . h Thus, we obtain that either σk is orthogonal to ϕk or the function ρ is found as [ ] c b − − + (n − 2)△1σ ρ = n 1 . n + 2 h Similarly, from (3.19) we obtain either σk is orthogonal to ψk or the function ρ is found as [ ] b c − − + (n − 2)△1σ ρ = n 1 . n + 2 So the proof is completed.

4. Examples

Let us consider a Riemannian metric g on the 4-dimensional real number space M 4 by [ ] 2 i j 1 2 2 2 3 2 4 2 ds = gijdx dx = (1 + 2p) (dx ) + (dx ) + (dx ) + (dx ) (4.1)

1 ex 1 2 3 4 where i, j = 1, 2, 3, 4, p = ρ2 and ρ is a non-zero constant and x , x , x , x are the standard coordinates of M 4. Then the only non-vanishing components of the Christoffel symbols, the curvature tensor, the Ricci tensor and scalar curvature are given by p Γ1 = − , Γ1 = Γ1 = −Γ1 = −Γ2 = −Γ3 = −Γ4 , 22 (1 + 2p) 33 44 11 12 13 14 p p R = R = R = ,S = 3 , 1221 1331 1441 (1 + 2p) 11 (1 + 2p)2 p p S = S = S = , r = 6 ≠ 0 , 22 33 44 (1 + 2p)2 (1 + 2p)3 272 s. dey, a. bhattacharyya and the components which can be obtained from these by the symmetry properties. Therefore M 4 is a Riemannian manifold (M 4, g) of non-vanishing scalar 4 curvature. We shall now show that M is a MG(QE)4, i.e., it satisfies (1.1). Let us now consider the associated scalars as follows: p 1 7 a = , b = − , c = −(1 + 2p)2 , d = . (4.2) (1 + 2p)3 (1 + 2p)4 4(1 + 2p) In terms of local coordinate system, let us consider the 1-forms ϕ and ψ as follows: {√ p(1 + 2p) for i = 1 , ϕi(x) = (4.3) 0 for otherwise , and  √  2 p 2 for i = 1 , ψi(x) = (1 + 2p) (4.4)  0 for otherwise , at any point x ∈ M 4. In terms of local coordinate system, the defining condition (1.1) of a MG(QE)4 can be written as

Sii = agii + bϕiϕj + cψiψj + 2dϕiψj . (4.5)

By virtue of (4.2), (4.3) and (4.4), it can be easily shown that (4.5) holds for 4 i, j = 1, 2, 3, 4. Therefore (M , g) is a MG(QE)4, which is not quasi-Einstein. Hence we can state the following: Let (M 4, g) be a Riemannian manifold endowed with the metric given in 4 (4.1). Then (M , g) is a MG(QE)4 with non-vanishing scalar curvature which is not quasi-Einstein.

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